Boolean unification with predicates

Eberhard, Sebastian; Hetzl, Stefan; Weller, Daniel S.

doi:10.1093/logcom/exv059

Cited by 6 publications

(8 citation statements)

References 18 publications

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“…The Boolean solution problem or Boolean unification with predicates is a computational task related to second-order quantifier elimination [44,42,56]. So far, PIE includes experimental implementations for special cases: Quantifier-free formulas with a technique from [16] and a version for finding solutions with respect to ground atoms, in analogy to the elimination of ground atoms.…”

Section: Results Of Interpolationmentioning

confidence: 99%

Facets of the PIE Environment for Proving, Interpolating and Eliminating on the Basis of First-Order Logic

Wernhard

2020

Preprint

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PIE is a Prolog-embedded environment for automated reasoning on the basis of first-order logic. Its main focus is on formulas, as constituents of complex formalizations that are structured through formula macros, and as outputs of reasoning tasks such as second-order quantifier elimination and Craig interpolation. It supports a workflow based on documents that intersperse macro definitions, invocations of reasoners, and L A T E X-formatted natural language text. Starting from various examples, the paper discusses features and application possibilities of PIE along with current limitations and issues for future research.

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Section: Results Of Interpolationmentioning

confidence: 99%

Facets of the PIE Environment for Proving, Interpolating and Eliminating on the Basis of First-Order Logic

Wernhard

2020

Preprint

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“…The Boolean solution problem or Boolean unification with predicates is a computational task related to second-order quantifier elimination [40,38,50]. So far, PIE includes experimental implementations for simple cases: Quantifier-free formulas with a technique from [13] and a version for finding solutions with respect to ground atoms, in analogy to the elimination of ground atoms.…”

Section: Results Of Eliminationmentioning

confidence: 99%

PIE -- Proving, Interpolating and Eliminating on the Basis of First-Order Logic

Wernhard¹

2019

Preprint

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PIE is a Prolog-embedded environment for automated reasoning on the basis of first-order logic. It includes a versatile formula macro system and supports the creation of documents that intersperse macro definitions, reasoner invocations and L A T E X-formatted natural language text. Invocation of various reasoners is supported: External provers as well as sub-systems of PIE , which include preprocessors, a Prologbased first-order prover, methods for Craig interpolation and methods for second-order quantifier elimination.

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“…A generalization of Boolean unification to predicate logic with various specific problems characterized by the involved formula classes has been investigated in [16]. The material presented here is largely orthogonal to that work, but a technique from [16] has been adapted to more general cases in Sect. 7.3.…”

Section: View As Unificationmentioning

confidence: 99%

“…Some of these relate to applications of Craig interpolation. Recent work by Eberhard, Hetzl and Weller [16] shows a constructive method for quantifier-free first-order formulas. A generalization of their technique to relational monadic formulas is shown, which, however, produces solutions that would be acceptable only under a relaxed notion of substitutibility.…”

Section: Approaching Constructive Solution Techniquesmentioning

confidence: 99%

“…A generalization of Boole's method beyond propositional logic to relational monadic formulas has been presented by Behmann in the early 1950s [6,7]. Recently the complexity of Boolean unification in a predicate logic setting has been investigated for some formula classes, in particular for quantifier-free first-order formulas [16]. A brief discussion of Boolean reasoning in comparison with predicate logic can be found in [8].…”

Section: Introductionmentioning

confidence: 99%

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The Boolean Solution Problem from the Perspective of Predicate Logic - Extended Version

Wernhard¹

2017

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Finding solution values for unknowns in Boolean equations was a principal reasoning mode in the Algebra of Logic of the 19th century. Schröder investigated it as Auflösungsproblem (solution problem). It is closely related to the modern notion of Boolean unification. Today it is commonly presented in an algebraic setting, but seems potentially useful also in knowledge representation based on predicate logic. We show that it can be modeled on the basis of first-order logic extended by secondorder quantification. A wealth of classical results transfers, foundations for algorithms unfold, and connections with second-order quantifier elimination and Craig interpolation show up. Although for first-order inputs the set of solutions is recursively enumerable, the development of constructive methods remains a challenge. We identify some cases that allow constructions, most of them based on Craig interpolation, and show a method to take vocabulary restrictions on solution components into account.

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Boolean unification with predicates

Cited by 6 publications

References 18 publications

Facets of the PIE Environment for Proving, Interpolating and Eliminating on the Basis of First-Order Logic

Facets of the PIE Environment for Proving, Interpolating and Eliminating on the Basis of First-Order Logic

PIE -- Proving, Interpolating and Eliminating on the Basis of First-Order Logic

The Boolean Solution Problem from the Perspective of Predicate Logic - Extended Version

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